Level Set Methods for Geometric Inverse Problems in Linear …frey/papers/elasticity... · 2017. 5. 9. · Level Set Methods for Geometric Inverse Problems in Linear Elasticity Hend

Level Set Methods for Geometric Inverse Problems in Linear

Elasticity

Hend Ben Ameur ∗

[email protected]

Martin Burger †

[email protected]

Benjamin Hackl‡

[email protected]

July 17, 2003

Abstract

In this paper we investigate the regularization and numerical solution of geometric inverseproblems related to linear elasticity with minimal assumptions on the geometry of the solution.In particular we consider the probably severely ill-posed reconstruction problem of a two-dimensional inclusion from a single boundary measurement.

In order to avoid parameterizations, which would introduce a-priori assumptions on thegeometric structure of the solution, we employ the level set method for the numerical solutionof the reconstruction problem. With this approach we construct an evolution of shapes witha normal velocity chosen in dependence of the shape derivative of the corresponding least-squares functional in order to guarantee its descent. Moreover, we analyze penalization byperimeter as a regularization method, based on recent results on the convergence of Neumannproblems and a generalization of Golab’s theorem.

The behavior of the level set method and of the regularization procedure in presenceof noise are tested in several numerical examples. It turns out that reconstructions of goodquality can be obtained only for simple shapes or for unreasonably small noise levels. However,it seems reasonable that the quality of reconstructions improves by using more than a singleboundary measurement, which is an interesting topic for future research.

Keywords: Geometric Inverse Problems, Linear Elasticity, Level Set Method, Regulariza-tion.

AMS Subject Classification: 35R30, 49Q10, 74B05, 65J20

1 Introduction

This paper is devoted to the regularization and numerical solution of some geometric inverse prob-lems in linear elasticity. We study the inverse problem of identifying interfaces or inclusions (withpossibly multiple connected components) from boundary measurements. We develop a rather gen-eral approach to such geometric inverse problems in linear elasticity, where the normal componentof the stress tensor satisfies a homogeneous boundary condition on the unknown geometry. The

∗SFB, ENIT, LAMSIN, BP 51, 1002 Tunis Belbedere, Tunisia. This work was started during the visit to theJK university (Linz) as a research fellow 2002.

†Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, LA, CA 90095-1555,USA. On leave from: Johannes Kepler Universitat Linz.

‡SBF F 013, Numerical and Symbolic Scientific Computing, Freistadterstr. 313, A-4040 Linz, Austria.§This work was supported by the Austrian National Science Foundation (FWF) under the grant SFB F 013 /

08 and P13478-INF

1

1 INTRODUCTION 2

boundary measurements in such applications are those of displacement on a part of the boundarywhere we have a boundary condition satisfied by the normal component of the stress tensor, too.

Geometric inverse problems, i.e., problems where the unknown is a geometric shape, are inves-tigated for around three decades (cf. e.g. Dervieux and Palmerio [12], Kohn and Vogelius[24]) with respect to theoretical and numerical aspects. A standard approach for the solution ofsuch problems consists in parameterizing the shape and applying regularization methods directlyto the parameterization (cf. e.g. Hettlich and Rundell [17, 18, 19]). This approach suffersfrom the limitation that a lot of a-priori knowledge on the structure and topology of the solu-tion shape has to been known in order to obtain convergent approximations. In particular, anyparameterization does not allow a change in the number of components, and hence, a shape canbe reconstructed if this number is known exactly. For the reasons described above, alternativeapproaches to the solution of shape reconstruction problems have been considered recently, suchas the sampling methods (cf. Kirsch [23]) or the level-set method (cf. Santosa [42]). Whilethe first one is based on properties of Dirichlet-Neumann type operators and therefore requires alarge number of measurements, the level set method can be applied also in the case of a singleDirichlet-Neumann measurement, which we therefore consider in this paper. The level set ap-proach was introduced by Osher and Sethian [37] for evolving geometries, with the originalaim of deriving fast algorithms for a flame propagation problem. The basic idea is to representa shape implicitly as the zero level set of a continuous function and to use the correspondence ofgeometric variations of the shape and the the solution of specific Hamilton-Jacobi equations forthe level set function (cf. Section 4 for further details). Due to the implicit representation on anEulerian grid, the level set approach does not introduce any a-priori assumptions on the geometryand therefore receives growing attention in the context of geometric inverse problems (cf. e.g.Burger [5, 6], Ito, Kunisch and Li [25], Ramananjaona et. al. [39, 40], Santosa et. al.[31, 42]). In this paper we shall investigate the application of level set methods in the context ofelastic inclusion detection, which is a challenging problem both theoretically and numerically dueto the Neumann condition on the boundary of the unknown shape.

By allowing rather general topology of the shape to be reconstructed, one also needs geometricregularization strategies independent of parameterizations. A frequently used approach is to adda multiple of the perimeter as a penalty to the least-squares functional. For problems that onlydependent on the shape via its indicator function, this approach is equivalent to total variation reg-ularization (cf. Rudin, Osher and Fatemi [41]) and can be analyzed in the standard frameworkdeveloped by Acar and Vogel [1]. For a problem with Neumann boundary condition on theunknown interface, the least-squares functional is not lower semi-continuous with respect to thestandard topologies for the indicator function and hence, a completely different approach has to beused. It turns out that the problem is lower-semicontinuous with respect to the Hausdorff-distanceof shapes if an appropriate weak formulation (in Deny-Lions spaces) of the Neumann-problem isused. In this paper we will prove that perimeter penalization is indeed a convergent regularizationmethod in this case, by using methods of geometric measure theory and novel results on the gen-eralization of Golab’s theorem (i.e., the lower semicontinuity of the perimeter with respect to theHausdorff-distance metric). Moreover, we investigate the behavior of this regularization strategyin several numerical examples and compare it to a direct regularization by a level-set evolution,where the regularizing effect comes from an early termination of the evolution.

Throughout this paper we consider a homogeneous isotropic linear elastic material, in a domainD ⊂ R3, assuming that there exists a surface Γ ⊂ D that separates the domain into two disjointopen sets D1 and D2, i.e.,

D = D1 ∪ D2, Γ = ∂D1 \ ∂D = ∂D2 \ ∂D.

The linear elastic problem under consideration is specified by

div σ(u) = 0 in D \ Γ = D1 ∪ D2

σ(u) . n = 0 on Γ (1.1)

supplemented by appropriate boundary conditions (specified below) on ∂Ω, n denotes a normalvector to Γ. In the following we will use the convention that this normal vector is in the outward

1 INTRODUCTION 3

normal direction of D1. The vector u denotes the displacement and σ(u) is the associated stresstensor, which is related via Hooke’s law to the linearized strain tensor ε(u) via

σ = λ trε I + 2µ ε . (1.2)

The linearized strain tensor ε(u) is given by

εij(u) =12

(∂ui

∂xj+

∂uj

∂xi

),

tr denotes the trace of a matrix, and λ, µ are Lame coefficients related to Young’s modulus E andthe Poisson ratio ν.

µ =E

2 (1 + ν)λ =

Eν

(1− 2ν) (1 + ν).

The original identification process can be stated as:

Inverse Problem 3D: Identify the unknown shape Γ by applying some prescribed load g onΓN ⊂ ∂D and measuring the displacement induced by g on some part ΓM ⊂ ΓN with ΓM havinga strictly positive measure.

We shall consider two two-dimensional cases derived from the three-dimensional problems forspecific geometrical situations and special applied loads, the so-called planar and anti-planar cases.In these situations, uniqueness and stability results can be shown under rather general geometricassumptions (cf. Ben Ameur, Burger and Hackl [3]), which provides a clear theoretical basis.The full three-dimensional case shall be considered in future research.

This paper is organized as follows: In the remaining part of the introduction we reduce the fullthree-dimensional problem (1.1) to two-dimensional problems, and formulate the inverse problemsthat will be investigated further in the following sections. The regularization of the identificationproblem with minimal assumptions on the regularity of the shape is discussed in Section 2. InSection 3, we derive a level set method to solve the regularized problems, respectively as a regu-larizing evolution itself. Moreover, we discuss the numerical solution of the elasticity problem bythe immersed interface method. We test the numerical behavior of the level set method and theregularization strategies in several numerical examples presented in Section 4, before we concludeand give an outlook to future work.

1.1 Planar and Anti-Planar Cases

We assume that D = Ω×R, where Ω is a bounded domain in R2, Ω = Ω1∪Ω2 and Σ = ∂Ω1∩∂Ω2.The different possible geometric situations are shown in Figure 1.

Furthermore we suppose that we apply a planar load g and that the displacement u : Ω×R −→R3 depends only on x1 and x2. We split the initial 3D problem (1.1) into two problems: A planarstrain one, where we consider u(x1, x2) = (u1(x1, x2), u2(x1, x2), 0), and an anti-planar problemdepending only on the third component u(x1, x2) = (0, 0, u3(x1, x2)).

In the planar strain case, we obtain a stress tensor σ satisfying σ3,` = σ`,3 = 0, ` = 1, 2. Thechoice of g with a third component equal to zero will allow us to obtain a two dimensional problemsimilar to the one corresponding to the plane stress case where σ3,` = σl,` = 0, l = 1, 2, 3. We referto both as “planar case” and rewrite (1.1) for the planar case as:

div σ(u) = 0 in Ω \ Σσ(u) . n = g on ΓN

σ(u) . n = 0 on Σu = 0 on ΓD

(1.3)

where ΓD, ΓN is a partition of the boundary of Ω supporting Dirichlet and Neumann boundaryconditions. The identification problem can then be stated as:

Inverse Problem, Planar Case: Identify the unknown interface Σ from a measurement ofthe displacement u on ΓM ⊂ ΓN (with ΓM having a strictly positive measure), where u = (u1, u2)

1 INTRODUCTION 4

is the solution of (1.3) with the constitutive law (1.2).

Ω

Ω

Σ1

2

ΓΓΝ

D

g

ΩΩ ΩΣ

Γ

22

1D

ΓN

g

Figure 1: Different situations

In the anti-planar case the original linear elasticity problem (1.1) reduces to a boundary valueproblem for a single Laplace equation:

µ ∆ u3 = 0 in Ω \ Σµ∂u3

∂n = g on ΓN

µ∂u3∂n = 0 on Σ

u3 = 0 on ΓD

(1.4)

where µ is a Lame constant.

For simplicity, we denote u3 by u in (1.4) when the anti-planar case is concerned. The asso-ciated identification problem as can then be formulated as:

Inverse Problem, Anti-Planar Case: Identify the unknown interface Σ from a measure-ment of the vertical displacement u on ΓM ⊂ ΓN (with ΓM having a strictly positive measure),where u is the solution of (1.4).

1.2 Weak Formulations of the Direct Problem

In this section we introduce weak formulations of the direct problems (1.3) and (1.4), respectively,under minimal assumptions on the regularity of the interface Σ. In order to allow for generalHausdorff-measurable interfaces Σ, we use a framework in Deny-Lions spaces, recently used forthe mathematical modeling and analysis of crack growth by Chambolle [8], Dal Maso andToader [32] and Giacomini [16]. For smooth interfaces Σ, this solution concept coincides withthe standard framework for weak solutions in Sobolev spaces.

In the following we assume that Σ is a compact subset of Ω with a finite number of connectedcomponents, which are measurable with respect to the Hausdorff measure H1(Σ) (we refer toFederer [15] and Morel, Solimini [33] for details on the definition and properties of Hausdorffmeasures). Under such general condition a standard weak formulation is not suitable anymore,since one cannot define Sobolev spaces like H1(Ω \ Σ) in a classical way. Therefore we introducea generalization of Sobolev spaces for non smooth domains as follows: If A is an open subset ofR2 (having in mind A = Ω \ Σ, the Deny-Lions space L1,2(A) is defined by

L1,2(A) := u ∈ W 1,2loc (A) | ∇u ∈ L2(A;R2) .

Note that for a regular A, the Deny-Lions space coincides with the usual Sobolev space H1(A),but it is a strict superset for non regular sets. However, since we assume that Σ is a compactsubset of Ω and ∂Ω is regular, each element in L1,2(Ω \ Σ) is embedded in a local H1-space in a

2 GEOMETRIC REGULARIZATION 5

neighborhood of the boundary so that we can define traces on ∂Ω in a standard way. Finally, wedefine the Deny-Lions space L1,2(A;R2) (cf. e.g. Giacomini [16] for further details).

L1,2(A;R2) := u ∈ W 1,2loc (A;R2) | ∇u ∈ L2(A;R2×2) .

In the planar case, we specify the space of admissible vertical displacements as

U := u ∈ L1,2(Ω \ Σ) | u|ΓD = 0 . (1.5)

The weak formulation of the anti-planar case consists in finding u ∈ U such that

a(u, v) :=∫

Ω\Σ

∇u.∇v dx =∫

ΓN

gv ds =: 〈G, v〉 ∀ v ∈ U . (1.6)

The solution of this problem is also the unique minimizer of the variational problem

a(u, u)− 2〈G, u〉 → minu∈U

.

In the planar case, we can rewrite the variational problem in an analogous way, now with thevectorial Deny-Lions space

U := u ∈ L1,2(Ω \ Σ;R2) | u|ΓD= 0 . (1.7)

and the variational form

a(u, v) :=∫

Ω\Σ

σ(u) : ε(v) dx =∫

ΓN

g.v ds =: 〈G, v〉 ∀ v ∈ U ,

with σ and ε defined as above, and σ : ε =∑

i,j σijεij .Finally we remark that for these weak formulations, existence and uniqueness of solutions for

given interface Σ is guaranteed by the results in Giacomini [16]. In particular, the shape-to-outputmap Σ 7→ u|ΓM

is well-defined, a fact that we shall use subsequently without further notice.

2 Geometric Regularization

Usually inclusion and interface identification problems are ill-posed. Even when for our problemclass a local stability result was proofed in Ben Ameur, Burger and Hackl [3] that mightindicate the well-posedness of the inverse problems under consideration, these stability results,however, are valid only under strong additional assumptions on the shape, and only yield localand directional information. In particular, they provide no information about the stability withrespect to general perturbations of the output data in L2(ΓM ), in whose norm we measure theerror. The somehow minimal requirements for well-posedness in this output space would meanthe existence of a solution to the least-squares problem

J0(Σ) = ‖u− θδ‖2L2(ΓM ) → minΣ∈K

in some appropriate class of shapes K together with weak stability of the minimizer with respect toperturbations of the data θδ. As observed from the following example, weak stability in a generalclass of shapes does not hold:

Example 2.1. Let Ω = [0, 1] × [−1, 1], ΓM = [0, 1] × 1 and define a sequence of interfaces Σn

viaΣn = (x1, x2) | x2 =

12

sin nπx1 with associated states un solving (1.4) and fn = un|ΓM . Moreover we pose homogeneous Dirichletconditions on ΓD = 0, 1 × (−1, 1) and some Neumann boundary conditions g on ΓN = [0, 1] ×


−1, 1, non-vanishing on ΓM only. If ψ is the unique solution of ∆ψ = 0 in Ω subject to thisboundary conditions, then by the definition of un we obtain that

∫

Ω

|∇un|2 dx + 2∫

ΓN

gun ds ≤∫

Ω

|∇ψ|2 dx + 2∫

ΓN

gψ ds := C

In particular we obtain for the integral over S = (0, 1)× ( 12 , 1)

∫

S

|∇un|2 dx + 2∫

ΓN

gun ds ≤ C.

The Poincare inequality gives for functions in H1(S) with homogeneous Dirichlet values on 0, 1×( 12 , 1) that the first integral is an equivalent norm to the H1-norm and if ‖g‖

H− 12 (ΓM )

is sufficientlysmall, we obtain that ∫

S

|∇un|2 dx ≤ C

2.

Thus, the restriction of un is uniformly bounded in H1(S) and due to the compactness of the traceoperator from H1(S) to L2(ΓM ) there exists a subsequence of fn converging to some f in L2(ΓM ),but the sequence Σn has no subsequence converging to a curve with finite Hausdorff measure H1.This example shows that without severe restrictions the problem is ill-posed even in any class ofΣ piecewise in Ck (for any k ∈ N) and H1-measurable, since the Σn constructed above are evenanalytic.

As usual for ill-posed problems we have to introduce a regularization approach in order tocompute stable approximations of the solution in presence of data noise (cf. Engl, Hanke andNeubauer [13] for an overview on this subject), In the following parts we consider two differentapproaches to the regularization of (1.1), (1.2). The first one is a penalization approach motivatedby the Mumford-Shah functional used frequently in image processing (cf. Morel and Solimini[33]), while the second consists in the early stopping of an evolution in artificial time, motivatedby asymptotic regularization of inverse problems.

2.1 Penalization by Perimeter

A common approach to the stabilization of inverse interface problems is to add the perimeter (i.e.,the Hausdorff measure H1) of Σ as a penalty to the output functional, i.e., to minimize

Jα(Σ) =12

∫

ΓM

|u− θδ|2 ds + αH1(Σ) → minΣ∈K

, (2.1)

where K denotes the class of compact subsets of Ω. Penalization by perimeter is well-analyzedin the context of the Mumford-Shah functional in image processing, where Σ represents the dis-continuity set of the function u, whose L2-distance to some given function is to be measured (seeMorel and Solimini [33] for a comprehensive overview). Recently, this strategy has been usedin the context of shape optimization or geometric inverse problems (cf. Ito, Kunisch and Li[21]). Problems with the analysis of this regularization strategy are caused in particular by the factthat the perimeter itself is not lower-semicontinuous in the class K with respect to the Hausdorffmetric, which seems to be natural if the convergence of shapes is considered.

The lower semicontinuity on the subclass K1 of simply connected, H1-measurable compact setsis provided by a classical result in geometric measure theory, called Golab’s Theorem (cf. Moreland Solimini [33]). Due to recent results by Dal Maso and Toader [32] and in particular byGiacomini [16], the lower-semicontinuity of the perimeter is guaranteed on the more interestingclass Km of H1-measurable compact sets with at most m connected components, which we shalluse in our analysis below:


Theorem 2.2. [32, Corollary 3.3] Let (Σn) be a sequence in Km converging to Σ in the Hausdorffmetric. Then Σ ∈ Km and

H1(Σ ∩ Ω) ≤ lim infn→∞

H1(Σn ∩ Ω).

Moreover, the results in these papers allow to deduce also the lower semicontinuity of the outputfunctional and thus the analysis of this regularization strategy in a similar way to the analysisof standard Tikhonov regularization for nonlinear inverse problems (cf. Seidman and Vogel[43], Engl, Kunisch and Neubauer [14]), respectively to the analysis of the the so-called totalvariation (TV) regularization (cf. Rudin, Osher and Fatemi [41], Acer and Vogel [1]). Theproof of the lower semicontinuity of the output-functional is the only property in this section forwhich we have to consider the two different cases of planar and anti-planar strain. We start withthe anti-planar one, for which we use the results of Dal Maso and Toader [32]:

Proposition 2.3 (Lower Semicontinuity in the Anti-Planar Case). The functional

J0 : Σ 7→∫

ΓM

|u− θ|2 ds

is lower semicontinuous on Km for arbitrary θ ∈ L2(ΓM ), i.e. for any sequence (Σn) in Km

converging to Σ in the Hausdorff metric, the solutions un respectively u of (1.5), (1.6) with Σreplaced by Σn respectively Σ, satisfy

∫

ΓM

|u− θ|2 ds ≤ lim infn→∞

∫

ΓM

|un − θ|2 ds.

Proof. Due to Dal Maso and Toader [32, Theorem 5.1], the gradients of the solutions un andu (extended by zero on Σn and Σ respectively) satisfy ∇un → ∇u strongly in L2(Ω;R2). Thus,for some open set U with ΓM ⊂ ∂U , we obtain that un → u in H1(U) and thus, a standard tracetheorem implies un|ΓM → u|ΓM in H

12 (ΓM ) → L2(ΓM ). Together with the lower semicontinuity

of the L2-norm this implies the assertion.

The proof in the planar case can be carried out in a completely analogous way, now using theresults of Giacomini [16, in particular Lemma 6.1] and will thus be omitted here:

Proposition 2.4 (Lower Semicontinuity in the Planar Case). The functional

J0 : Σ 7→∫

ΓM

|u− θ|2 ds

is lower semicontinuous on Km for arbitrary θ ∈ L2(ΓM ;R2), i.e., for any sequence (Σn) in Km

converging to Σ in the Hausdorff metric, the solutions un respectively u of (1.2),(1.7) with Σreplaced by Σn respectively Σ, satisfy

∫

ΓM


∫

ΓM

|un − θ|2 ds.

In the following we will not distinguish the planar and the anti-planar case, by consideringdata f ∈ L2(ΓM ) we will denote either scalar- or vector-valued functions. Moreover, for each Σwith some index, the function u, with the same index, will either denote the solution of (1.5),(1.6)or (1.2),(1.7), which we will use without further notice.

The first two results are concerned with the existence and (set-valued) stability of minimizersof (2.1) for α > 0:

Proposition 2.5 (Existence of a Minimizer). For any θδ ∈ L2(ΓM ) and any α > 0 thereexists a minimizer Σδ

α ∈ Km of (2.1).


Proof. Let (Σn) be a minimizing sequence in Km (which exists since the functional is boundedbelow by zero), then due to the boundedness of Ω, there exists a convergent subsequence in theHausdorff metric dH with limit Σδ

α ∈ Km. From the lower semicontinuity results in the precedingpropositions we deduce that

12

∫

ΓM

|uδα − θδ|2 ds + αH1(Σδ

α) ≤ 12

∫

ΓM

|un − θδ|2 ds + αH1(Σn),

and thus, Σδα is a solution of (2.1).

Proposition 2.6 (Stability). Let α > 0, let (θn) be a sequence in L2(ΓM ) converging to θδ andlet (Σn) the associated sequence of minimizers in Km of (2.1) with data θn. Then, there existsa subsequence of (Σn) convergent in the Hausdorff-metric dH and the limit of each convergentsubsequence is a solution of (2.1) with data θδ.

Proof. First of all, the existence of the minimizers Σn ∈ Km is guaranteed by Proposition 2.5.The existence of a convergent subsequence in the Hausdorff metric dH follows from Σn ⊂ Ω andthe boundedness of Ω. Now let Σn denote a converging subsequence with limit Σδ

α and let un,uδ

α denote the associated sequence of states. Then, from the strong convergence θn → θδ and thedefinition of un and Σn we deduce the inequality

12

∫

ΓM

|uδα − θδ|2 ds + αH1(Σδ

α) ≤ lim infn→∞

12

∫

ΓM

|un − θδ|2 ds + αH1(Σn)

≤ lim infn→∞

12

∫

ΓM

|un − θn|2 ds + αH1(Σn)

≤ limn→∞

12

∫

ΓM

|u− θn|2 ds + αH1(Σ)

≤ 12

∫

ΓM

|u− θδ|2 ds + αH1(Σ)

for any Σ ∈ Km with associated state u. Hence, the limit Σδα is also a solution of problem (2.1).

Now, we are able to prove the main result of this section, namely that penalization by perimeteris a convergent regularization method in Km equipped with the Hausdorff metric dH :

Theorem 2.7 (Convergence). Suppose that there exists Σ∗ ∈ Km such that the accordingsolution u∗ of (1.5),(1.6) or (1.2),(1.7), respectively, satisfies u∗|ΓM = θ. Let θδ be such that

‖u∗ − θδ‖L2(ΓM ) ≤ δ (2.2)

holds and let α be chosen such that

α → 0,δ2

α→ 0 as δ → 0.

Then, for the according solutions Σδα of (2.1), there exists a subsequence convergent in the Haus-

dorff metric dH and the limit Σ of every convergent sequence is a minimum-perimeter solution ofthe identification problem, i.e. the associated displacement u satisfies u|ΓM = θ and for all othersolutions Σ, we have that

H1(Σ) ≤ H1(Σ).

Proof. Again by boundedness, there exists a subsequence (Σn) := (Σδnαn

) converging in the Haus-dorff metric dH . Now let (Σn) be such a convergent subsequence with limit Σ. By the definitionof Σn we obtain by comparison with Σ∗ that

12

∫

ΓM

|un − θδn |2 ds + αnH1(Σn) ≤ 12

∫

ΓM

|θ − θδn |2 ds + αnH1(Σ∗) =δ2n

2+ αnH1(Σ∗).

3 NUMERICAL SOLUTION 9

Division by αn and the assumption δ2n

αn→ 0 shows that H1(Σn) is uniformly bounded and the

lower semicontinuity of the Hausdorff measure yields H1(Σ) ≤ H1(Σ∗). Moreover, as δn → 0, weobtain that

12

∫

ΓM


12

∫

ΓM

|un − θδn |2 ds ≤ lim infn→∞

δ2n

2+ αnH1(Σ∗) = 0.

Thus, Σ is a solution of the identification problem and since the solution Σ∗ is arbitrary, it is alsoa minimum-perimeter solution.

We finally remark that we obtain strong convergence by a standard principle, if the minimum-perimeter solution is unique. In particular this would be guaranteed by an identifiability result inthe class Km, but so far we can only give such a result for disjoint unions of smooth curves (cf.Ben Ameur, Burger, Hackl [3]).

3 Numerical Solution

In this section we discuss a level set approach to the solution of the identification problem, respec-tively to the optimization problem (2.1) obtained from penalization by perimeter. Furthermore wediscuss the solution of the arising Hamilton-Jacobi equation as well as the solution of the ellipticproblem with moving interface.

3.1 Level Set Methods

The level set approach to geometric motion has been introduced by Osher and Sethian [37].Its main idea is to represent an evolving front as the zero level set of a continuous function, i.e.

Σ(t) = x ∈ Ω | φ(x, t) = 0 .

A weak formulation of geometric motion with normal speed Vn is given by the Hamilton-Jacobiequation

∂φ

∂t+ Vn|∇φ| = 0 in Rd × R+ , (3.1)

in the sense that a viscosity solution for this Hamilton-Jacobi equation (3.1) has to be computed.We refer to the monograph by Lions [30] and the paper by Crandall et al. [10] for detailson the notion of viscosity solution. The function Vn in the Hamilton-Jacobi equation (3.1) is anextension of the velocity from the front to the whole RN (or subset of RN being the computationaldomain and including the interface).

The basic idea of level set methods for inverse and optimization problems (cf. Burger [5],Ito, Kunisch and Li [21], Santosa et al. [36, 42] and Ramananjaona et al. [39, 40])is to choose the velocity in such a way that a decrease of the objective functional is achieved,which resembles the classical speed method in shape optimization (cf. Murat and Simon [34],Sokolowski and Zolesio [46]), but the weak formulation via the level set method allows formore general evolutions and in particular for topological changes such as splitting or merging ofdomains. We restrict our attention again to the anti-planar case, noting that analogous reasoningis possible for the planar case.

In the following we discuss the construction of level set methods based on the original sharpinterface model for the state equation. The basic tool for the choice of the velocity is the shapederivative in direction of a normal velocity Vn (cf. Sokolowski and Zolesio [46] and Delfourand Zolesio [11]), which coincides with the time derivative of the objective functional Jα, whenthe shape is evolved according to the geometric motion defined by (3.1). For the detailed compu-tation of shape derivatives we refer to Ben Ameur, Burger, Hackl [3], here we briefly present


the result in the anti-planar case, the planar case can be treated in an analogous way. The timederivative of J(Σ(t)) under a geometric motion with normal velocity is given by

d

dtJα(Σ(t)) =

∫

ΓM

(u− θδ) u′ ds + α

∫

Σ

Vnκ dH1, (3.2)

where κ denotes the mean curvature of Σ, and u′ is the solution of

∆ u′ = 0 in Ω \ Σ∂u′∂n = 0 on ΓN∂u′∂n = −divT (Vn∇T u) on Σu′ = ∂u

∂n Vn on ΓD

(3.3)

Introducing the adjoint problem

∆ w = 0 in Ω \ Σ∂w∂n = χΓM

(u− θδ) on ΓN∂w∂n = 0 on Σw = 0 on ΓD

(3.4)

where χΓM is the indicator function of ΓM , we can deduce a simple formula for the derivative ofJα:

d

dtJα(Σ(t)) =

∫

Σ

(−∇T u.∇T w + ακ)Vn ds (3.5)

The Hadamard speed method consists in choosing the velocity Vn such that a steepest descent(or a gradient flow) is achieved with respect to an appropriate norm for the velocity Vn (cf.Sokolowski and Zolesio [46], Burger [6]). As discussed in Burger [6] we have severalpossibilities to choose the velocity via the speed method, in particular we consider the steepest inL2(Σ(t)) and in H

12 (Σ(t)). For the L2-case we obtain the velocity on Σ(t) as

Vn = [∇u.∇w]− ακ := V 0n − ακ, on Σ(t),

which corresponds to the classical version of the speed method and the standard approach for levelset methods in inverse obstacle problems. Since preliminary numerical experiments indicate thatthis standard approach fails for the problem we consider, we use the gradient flow in the norm ofH

12 (Σ(t)), which lead to reasonable results in Burger [6]. In the case of the H

12 -norm, we have

additional smoothing of the velocity Vn via

∆ ψVn = 0 in Ω \ Σ∂ψVn

∂n = 0 on ΓN∂ψVn

∂n = [∇u.∇w]− ακ on Σ(t)ψVn = 0 on ΓD

(3.6)

and then obtain the velocity as the trace Vn = ψVn on Σ(t). Since ψVn is an extension of Vn tothe whole domain Ω, we can use it directly as an extension velocity for the level set method, sothat the resulting evolution equation is given by

∂φ

∂t+ ψVn |∇φ| = 0.

Since the normal jump has a meaning only on Σ(t) we need to compute an extension velocity atleast in a neighborhood of Σ(t), which can be realized e.g. by solving a stationary Hamilton-Jacobiequation as in Adalseinsson and Sethian [2]. Alternatively, we could also perform a strategyas for the H

12 -case and solve the problem (3.6) with Dirichlet data V 0

n . Extensions of th curvature


to the whole domain can be obtained as (cf. Osher and Sethain [37]) κ = div(∇φ|∇φ|

), which

finally leads to the level set evolution (with V 0 being an appropriate extension of V 0n )

∂φ

∂t+ V 0|∇φ| − α|∇φ| div

( ∇φ

|∇φ|)

= 0.

For α = 0, this evolution is a first-order Hamilton-Jacobi equation with a non local term causedby the dependence of V 0 on the zero level set; for α > 0 the evolution is of second order.

We want to mention that in the case α = 0, the associated least-squares problem is ill-posed,but we may still consider the speed method as an evolutive regularization method. As for anyregularizing evolution for ill-posed problems in presence of noise, an appropriate stopping criterionis needed relating the time T∗ at which the evolution is terminated with the noise level δ in (2.2).By analogy to the method of asymptotic regularization (cf. Tautenhan [47]) and to a regularizinglevel set evolution for a class of shape reconstruction problems (cf. Burger [5] where a rigorousanalysis has been carried out), we propose to use the discrepancy principle, choosing T∗ via

T∗(δ, θδ) := inf t ∈ R+ | ‖u(t)|ΓM− θδ‖L2(ΓM ) ≤ τδ

for appropriate (fixed) parameter τ > 1. We shall compare the results obtained with this regular-ization approach, which we call terminated speed method, to the ones obtained with penalizationby perimeter in the next section.

3.2 Solving the Hamilton-Jacobi equation (3.1)

For solving the Hamilton-Jacobi equation (3.1) numerically, we merely use well-known schemes.There is a vast literature on numerical solvers for Hamilton-Jacobi equations (e.g. see see Osheret al. [35, 38], Jiang and Peng [22]) being based on the idea of essentially non-oscillatoryschemes.

In all our numerical tests we used a 5th order WENO scheme to discretize the spatial partof the Hamilton-Jacobi equation and an explicite Runge-Kutta scheme of 3rd order to solve thetime part of the Hamilton-Jacobi equation (see Jiang and Peng [22]). Due to the finite domainwe added homogeneous Neumann Boundary conditions to the Hamilton-Jacobi equation. Theinfluence of this boundary condition can be ignored however, since the inclusions do not touch theboundary in the cases under consideration.

The explicit time integration of the Hamilton-Jacobi equation requires to fulfill the CFL con-dition to maintain a stable Hamilton-Jacobi solver. This means that the time step ∆t is restrictedby

∆t ≤ ch

‖ψVn‖L∞(Ω),

with a constant c < 1, where h denotes the mesh size. In all our numerical tests we chose c = 0.8,which produced reasonable results.

According to theory the calculated velocity Vn should result into a descent of the objectiveJα(Σ) (equation (2.1)). By using the CFL condition to determine the time step in the Hamilton-Jacobi solver we observed that this might also result in an increase of the objective. This maycaused by the fact that the CFL-time step is too large or due to a poor approximation of thesolution of the PDE (1.4) or the adjoint PDE (3.4). Note that it may happen that one obtainsa high approximation error for the PDE (1.4) and the adjoint PDE (3.4) because of a poorapproximation of the normal to the zero level set, which occurs if the level set function is highlyoscillating close to the zero level set. In this case a reinitialization of the level set function mayhelp.

Obviously, the increase of the objective Jα(Σ) during the iteration is not desirable in anoptimization context and hence, additionally to the CFL restriction, we restricted the time step∆t such that a decrease in the objective is guaranteed. This was realized by implementing a simpleline-search (bisection), up to some minimal step size (at most 10 bisections), which nonetheless


caused some increases in the objective. Due to the possible poor approximation of the normal atthe zero level set we additionally reinitialized the level set function to the signed distance functionwhen the step size reached its minimum. Nonetheless, the average behavior of the objective Jα(Σ)is decreasing so that the remaining increases in the objective can be accepted.

Finally we want to remark on the termination criteria for the perimeter regularization. Usuallyone would use a standard termination criteria from optimization for the regularized problem, e.g.,terminate when

dH(Σi+1, Σi) ≤ ε

where ε is a small positive number. Our calculation of the Hausdorff distance has accuracydH = O(h), so it is impossible to choose ε very small (ε < h). On the other hand the choice ε ≥ hwould cause a too early termination of the algorithm. Hence, we chose to terminate according toa proper chosen number of iterations such that by experience the Hausdorff distance dH did notchange.

3.3 Solving the Elliptic PDE (1.4)

One of the major issues in the numerical solution of inverse obstacle problems, crack identification,shape optimization and also moving boundary problems is the type of discretization used for theunderlying elliptic or parabolic partial differential equations, in particular the question whetherthe mesh should be changed during the optimization or evolution process or not. Mainly threedifferent approaches to this problem can be found in literature, namely finite element method,fictitious domain methods, and the immersed interface method. For our numerical simulations,we used the latter, mainly because of the theoretically well studied properties, the fixed grid forall time (optimization) steps and due to the available Fortran sources for the IIM solver, providedby Zhilin Li. One further advantage of the IIM in our application is that the finite differencegrid can also be used to discretize the Hamilton-Jacobi equation (3.1).

All standard finite element discretization of the PDE (1.4) face one problem in the level-setapproach, namely the need of two different meshes, a triangular one for the elliptic equationand a rectangular finite difference grid for the Hamilton-Jacobi equation. This is mainly becausealmost all well analyzed discretization schemes for the Hamilton-Jacobi equation are based onfinite difference discretization. It is usually not desirable to map quantities from one grid toanother, which may introduce a loss in accuracy and further numerical instabilities. However, wewant to remark that numerical tests with a finite-element based fictitious domain method lead tosimilar results as for the IIM.

The immersed interface method was originally developed to treat elliptic interface problemswith a finite difference discretization. It is proved to be of optimal accuracy, furthermore theinterface and the finite difference grid are independent. For more details see Li et al. [26, 27, 20,29]. The IIM seems to be quite well suited for interface identification problems because the finitedifference grid for the PDE can be fixed once and for ever (no remeshing). Using a similar trickas in the fictitious domain method it is also possible to treat inclusion or boundary identificationproblems with the IIM. This procedure is described in more detail in the following Section 3.3.1.

3.3.1 Immersed Interface Method for Inclusions

Originally the IIM was developed to treat interface problems with a finite difference discretization(see Li et al. [26, 20, 29]). To solve also inclusion or boundary problems (1.4) with the IIMwe have to transform the inclusion or boundary problem to an interface problem. We do thistransformation along the suggestions of Li et al. [27, 21] for the inclusion problem (1.4) only.

First we extend the solution of the inclusion problem (1.4) to the inclusion Ω− such that

∆u = 0 in Ω− .

For this sake we can either use the extension in H1(Ω), i.e. [u]|Σ := u+ − u− |Σ = 0, or requirethe jump of the normal derivative

[∂u∂n

] |Σ at the inclusion to be zero. In this way we get naturally


the following equivalent formulations of the inclusion problem (1.4).

∆u =0 in Ω+

u|ΓD =0∂u∂n |ΓN

= f∂u∂n |Σ =0

⇔

∆u =0 inΩ \ Σu|ΓD

= g∂u∂n |ΓN

= f[u]|Σ =0[ ∂u∂n ]|Σ =− ∂u

∂n−

⇔

∆u =0 inΩ \ Σu|ΓD

= g∂u∂n |ΓN

= f[u]|Σ = u+

[ ∂u∂n ]|Σ =0

(3.7)

Solving any of these equivalent formulations (if meas(ΓD) > 0) is a well posed problem (inparticular unique). The last two formulations are interface problems in a fixed point form.

Introducing a new variable w for either − ∂u∂n−|Σ or u+|Σ we get a Saddle point form that is

equivalent to the original PDE (3.7)

∆u =0 in Ω \ Σu|ΓD

= g∂u∂n |ΓN

= f[u]|Σ =0[ ∂u∂n ]|Σ − w =0

∂u∂n+

|Σ =0

⇔

∆u =0 in Ω \ Σu|ΓD

= g∂u∂n |ΓN

= f[u]|Σ − w =[ ∂u∂n ]|Σ =0

u−|Σ =0

Note that the restriction ∂u∂n+

|Σ = 0 and u−|Σ = 0 has to be done in H− 12 (Σ) and H

12 (Σ).

Next we discretize this system according to the IIM, i.e., the interface problem becomes

Au + Bw = F ,

where A is the discretization of the Laplacian (e.g. 5-point stencil) and B, F is calculated due tothe rules for IIM (see Li [27]). Like in the work of Ito, Li and Kunisch [21, 27] we discretizethe restriction ∂u

∂n+|Σ, u+|Σ = 0 using some properties of the IIM resulting into

Cu + Dw = G ,

where C,D, G are matrices respectively vectors obtained from the immersed interface discretiza-tion. One requirement we do not meet, using this discretization of the restriction, is that wediscretize the restriction in L2(Σ) (more accurate l2(Σ)) instead of H

12 (Σ) or H− 1

2 (Σ). Dueto tests done by Li [27] this seems not to have much influence and so we nonetheless use thisdiscretization.

Finally we can eliminate the variable u by using the inverse of the operator A. Computationallythe application of this inversion can be realized very efficiently by fast Poison solvers. When wesubstitute u into the equation for the restriction we end up with a smaller Schur complementequation

(D − CA−1B)w = G− CA−1F .

We solve this equation by the generalized minimum residual method (GMRES). According toLi [27] the number of iterations to solve this equation with the GMRES is independent of themesh size (which we also observed numerically for the inclusion problem) and so we obtain anefficient method to solve the inclusion problem (3.7). Clearly some analysis is missing, especiallyconcerning the stability of the discrete system, i.e., whether the Schur complement can be invertedstable and independent of the mesh size, or not.

Remark 1.

• The operators A,A−1, B, C, D are never calculated explicitely. Only their applications areneeded, which makes computations efficient.

• For certain shapes, especially shapes with very high curvatures (locally), we observed thatthe solution to the discrete system do not fit to the real solution very well. Probably this iseither due to an instable calculation of the operators B, C, D and vectors F, G according tothe IIM or due to a not properly converging GMRES procedure caused by the wrong spacefor the restriction equation (l2(Σ) instead of H

12 (Σ), H− 1

2 (Σ)).

4 NUMERICAL RESULTS 14

• In our calculations we used the reformulated PDE (3.7) with the Dirichlet restriction

u−|Σ = 0.

4 Numerical Results

In the following we shall report on some numerical experiments carried out with the methodsdescribed in the previous section to test the newly developed theory about the perimeter regular-ization as well as the proposed terminated speed method.

We restrict ourselves to the anti-planar case (1.4), because numerical tools are available forthis case, similar results can be expected for the planar case. All performed tests are done withthe same test configuration, i.e. we did all calculations on a square domain Ω = [−1, 1] × [−1, 1]with fixed boundary conditions. Only the inclusion Ω− and the corresponding measurements varyfrom example to example. The geometric configuration, the boundary conditions and the problemspecification are given in the following picture:

(-1,-1) (1,-1)

(1,1)(-1,1)

∂u∂n = −5 sin(4πx)

∂u∂n = 10

∂u

∂n

=10

cos(

3πy)

u=

0ΣΩ−

Ω+

∆u =0 in Ω+

u|ΓD=0

∂u∂n |ΓN

= h∂u∂n |Σ =0

Measurements:

θδ = u|ΓNin L2(ΓN )

For the starting value φ0 of the Hamilton-Jacobi equation (3.1) we chose for all test examplesφ0 = x2 + y2 − 0.32, i.e. a level set function having as zero level line a circle with center (0, 0)and radius 0.3.

To avoid inverse crimes (see comments by Colton and Kress [9, p. 133]) we calculated ourmeasurements on a very fine grid (e.g. 2049x2049), having no grid points in common with the onelater used for the numerical solutions of the inverse problem. These artificial measurements wereprojected by linear interpolation to the grid for the inverse problem.

The stability and convergence of the terminated speed method as well as the perimeter reg-ularization was tested by perturbing the measurements with some artificial noise (of levels δ =5, 3, 1, 0.8, 0.5, 0.3, 0.1%). We generated this noise using Gaussian random values for every mea-surement point. Then we scaled this artificially produced noise by its L2(ΓN )-norm, which wasadded, scaled by its (normalized) noise level δ‖u‖ΓN

100 , to the (exact) measurements.

Remark 2. Even for exact measurements our solver faces noisy data. This is due to the (coarse)discretization of the PDE. For the discretization we chose we expect approximately an additionalnoise of ∼ 0.1%.

Terminated Speed Method: The discrepancy principle was chosen to terminate with τ = 2,i.e. solve the Hamilton-Jacobi equation (3.1) as long as ‖u(T )− θδ‖L2(ΓM ) ≥ 2δ‖θδ‖L2(ΓN ) .

Perimeter Regularization: For the perimeter regularization we chose the penalization param-eter α to be proportional to the noise level, i.e., α = cδ. According to the theory developedin Section 2.1 this is a stable and converging method. Since we do not have a useful rulehow to chose the constant c, we used c = 5 for all our test examples (corresponding merelyto the scaling of the problem).


For almost all tests we plotted the Hausdorff distance dH(Σδα,Σ∗) versus the the noise level

δ (in logarithmic scale). Additional to the Hausdorff distance which is a very strong measure tocompare the quality of several identification results, we also plotted the L1 distance, i.e.

d1(Σδα,Σ∗) =

∫

Ω

|χφδα≤0 − χφ≤0|dx ,

versus the noise level δ (logarithmic scale).For inverse problems there are usually no benchmark examples available. Hence we chose our

test examples such that they are somehow representative for this class of problems, not too simpleand challenging for the algorithm.

4.1 Ellipse

For the first test example the inclusion is an ellipse with respect ratio 5/3.

Σ∗ =

(x, y) |

( x

0.5

)2

+(

y − 0.10.3

)2

= 1

This is probably the simplest example so we can expect that the algorithm performs well. Forthe discretization of the PDE (3.7) and the Hamilton-Jacobi equation (3.1) we chose a 257x257grid. In Figure 2 we present the calculated inclusions for the noise levels δ = 5, 1, 0.5, 0.1% as well

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(a) Terminated Speed Method

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(b) Perimeter Regularization

Figure 2: Ellipse: Identified inclusion for δ = 1, 0.5, 0.1%

as the exact shape for both the terminated speed method and the perimeter regularization. Oneobserves that the reconstructed inclusion approaches the exact inclusion with decreasing noiselevel, as predicted by theory. Only parts of the inclusion close to the Dirichlet boundary andclose to the Neumann boundary with constant load are of poor quality. For the part close tothe Dirichlet boundary this is not unexpected, since here we have no measurements and so theobjective will not be very sensitive to changes close to this boundary. In Figure 3 we plotted theHausdorff distance dH respectively the L1-distance d1 with respect to the noise level δ. The figureconfirms, what we already recognized in Figure 2, that the quality of the identified inclusion Σδ

α

improves with decreasing noise level. Both measures, the Hausdorff distance and the L1-distance,are decreasing with the noise level δ.


0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.1 1 10

d H(Σ

δ α, Σ

∗ )

δ

Terminated Speed MethodPerimeter Regularization

(a) Hausdorff distance dH(Σδα, Σ∗)

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.1 1 10

d 1(Σ

δ α, Σ

∗ )

δ


(b) L1-distance d1(Σδα, Σ∗)

Figure 3: Ellipse: Convergence in Hausdorff- and L1-distance versus noise δ

4.2 Sharp Ellipse

In the second test example the inclusion is as well an ellipse but with large respect ratio 6.

Σ∗ =

(x, y) |

( x

0.6

)2

+(

y − 0.10.1

)2

= 1

The large respect ratio causes a rather large curvature which is a challenge for the PDE solver.Also the inverse solver suffers from this large curvature. This time we discretize the PDE (3.7)and the Hamilton-Jacobi equation (3.1) by a 513x513 grid in order to obtain a better resolutionof the geometry. Only noise levels above δ ≥ 1% were done on a 257x257 grid.

Note that due to the large respect ratio of the ellipse for a 257x257 grid there are only a fewgrid points in the inclusion, which is a bit dangerous for the immersed interface discretization. As

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Figure 4: Sharp Ellipse: Identified inclusion for δ = 5, 1, 0.1%

before we present in Figure 4 the reconstructed inclusions for the noise levels δ = 5, 1, 0.1% as wellas the exact shape Σ∗ for both, the terminated speed method and the perimeter regularization.Also here the reconstruction improves with decreasing noise. Nonetheless the reconstruction is


still of poor quality for the finest noise level. The length and the width of the ellipse is almostcorrect but the curvature at the sharp radius is completely far off for the terminated speed methodwhereas the identified shape by the perimeter regularization gets too narrow. One may expectthat for high curvature the reconstructed shape will not be very accurate, but surprisingly now thepart of the interface with small curvature that is close to the Neumann boundary with constantload is far off the exact solution. For the same reason as for the ellipse with moderate respectratio the reconstruction close to the Dirichlet boundary is of poor quality.

0.12

0.14

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0.18

0.2

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0.3

0.32

0.1 1 10

d H(Σ

δ α, Σ

∗ )

δ



0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.1 1 10

d 1(Σ

δ α, Σ

∗ )

δ



Figure 5: Sharp Ellipse: Convergence in Hausdorff- and L1-distance versus noise δ

Despite the poor quality of the reconstruction in Figure 4 the error of the reconstructions in theHausdorff distance decreases, as seen in Figure 5, with the noise level. This is again in accordancewith our theory about the perimeter regularization and confirms the terminated speed method.For the perimeter regularization the L1-distance seems to increase but this is not in contradictionto the theory developed before which is only valid for the Hausdorff distance dH .

4.3 Star Shaped

Up to now all test examples where convex which is usually a quite nice property. In order to testa non-convex shape our third example is a star shaped inclusion.

Σ =(x, y) | x2 + (y − 0.1)2 =

(0.2 + 0.1 ∗ exp1.4∗cos(3∗κ−2)+0.4∗sin κ

)2,

κ = arctan xy−0.1

The star shaped inclusion itself has again a quite large curvature and additionally the curvature ischanging sign. So we can expect that this test example is quite hard for the algorithm. Especiallyfrom the last two examples we can not expect reconstructions of high quality in presence of noise.

For the discretization of the PDE (3.7) and the Hamilton-Jacobi equation (3.1) we chose a257x257 grid.

As expected we observe in Figure 6 that the reconstructions get very poor, for both theperimeter regularization and the terminated speed method. Even worse Figure 7 indicates thatthe perimeter regularization is not converging which would contradict the theory. Fortunately weare so far off the solution that we can argue that the result do not contradict the theory which isonly a local result. Despite this negative result one must be aware of the fact that the inclusion


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Figure 6: Star Shaped: Identified inclusion for δ = 5, 1, 0.1%

detection problem from a single boundary measurement is probably severely ill-posed, so we couldnot expect much better reconstructions.

We finally remark that even for visually poor reconstructions (cf. Figure 6) and increasingHausdorff- and L1-distance with decreasing noise level, the reconstruction is improved locallyat the three arms of the star. The curvature of the reconstructed inclusion starts to changesign, too, when the noise is small enough. Hence, one would expect much better reconstructionsfor significantly lower noise levels, which seems to be unrealistic in practice. The only way toimprove the quality of the reconstruction seems to be the use of multiple measurements (i.e., formeasurements for different Neumann values). The application of level set methods in this case isan interesting and important topic for future research.

4.4 Two Circles

Our final example is probably the most challenging one. The inclusion consists of two circles,which can in principle be handled by the level set method as opposed to most classical shapereconstruction algorithms based on parameterizations. Again the PDE (3.7) and the Hamilton-Jacobi equation (3.1) are approximated by a 257x257 grid.

Σ = (x, y) | (x + 0.4)2 + (y − 0.4)2 = 0.32 or(x− 0.3)2 + (y + 0.3)2 = 0.42

The identifiability result in Ben Ameur, Burger and Hackl [3] would theoretically allow toidentify these multiply connected inclusions uniquely, but from the numerical tests in the previousexamples it seems doubtful if this is possible in presence of noise. Indeed, the numerical results(cf. Figure 8) show that the initial domain does not split, but the algorithm shows at least acorrect behavior, i.e., the Hausdorff distance between exact and reconstructed shape is decreasingwith δ, and the shape seems to converge toward the correct inclusion.

For the terminated speed method we were not able to terminate according the terminationcriteria for small noise levels δ, it seems that the evolution got stuck in a local minimum. Thisindicates that also the reconstruction obtained with the perimeter regularization (looking similarly)might be a local minimum and not a global one. Figure 8 presents the calculated inclusions for

5 CONCLUSIONS AND OPEN PROBLEMS 19

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.1 1 10

d H(Σ

δ α, Σ

∗ )

δ


(a) Hausdorff distance dH(Σδα, Σ)

0.2

0.25

0.3

0.35

0.4

0.45

0.1 1 10

d 1(Σ

δ α, Σ

∗ )

δ


(b) L1-distance d1(Σδα, Σ)

Figure 7: Star Shaped: Convergence in Hausdorff- and L1-distance versus noise δ

noise levels δ = 1, 0.5, 0.1 % as well as the exact shape for the perimeter regularization. Visually,the reconstructions seem to get better when the noise δ reduces but still the components do notsplit into two components. As for the star shaped problem (Section 4.3) Figure 9 indicates thatthe Hausdorff distance for the perimeter regularization is increasing with the noise. This is eithercaused by getting stuck in a local minima or since the reconstructions are just too far from theexact solution to allow the application of the theoretical results.

5 Conclusions and Open Problems

We have presented a general approach to geometric inverse problems in linear elasticity, for whichwe also provided a convergent regularization method under very general geometric assumptions.Moreover, we have discussed the numerical solution of these inverse problems using the level setmethod, which allows to consider general shapes and does not rely on any parameterization.

The numerical results obtained for the anti-planar case are encouraging for future research.Especially an extension to more than one set of boundary measurements seems to be interesting.From related problems like impedance tomography and inverse scattering with full measurements(Dirichlet to Neumann map) it is well-known that the two-dimensional problem is not overde-termined, whereas the three-dimensional one is. Hence, a possible application of the inclusiondetection problem to the general three-dimensional case (with full measurements) , which also isof importance in practice, may produce better results.

For the terminated speed method an analysis is still missing but the numerical results comparedto the yet theoretically analyzed perimeter regularization give a good evidence to proceed withanalyzing this method.

Moreover, many theoretical questions related to the solution by the level set method are stillopen, such as the well-posedness of this evolution and its regularizing properties. In particularfor regular shapes there is some hope that we might apply local stability results obtained in BenAmeur, Burger and Hackl [3] to obtain a convergence rate of the level set evolution, since theevolution uses exactly a path determined by the shape derivative, to which the stability estimatescan be applied.

Analysis and computation is still open in the three-dimensional case, but we hope to obtain

REFERENCES 20

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Figure 8: Two circle: Identified inclusion for δ = 1, 0.5, 0.1%

stability results in a similar way as for the two-dimensional planar case. Moreover, analogousquestions are still open for thermo-elasticity, where the structural mechanics problems is coupledto a heat equation or for other types of material laws.

Acknowledgments

We thank Prof. Heinz W. Engl (University Linz) for stimulating this collaboration and for usefuldiscussion. Moreover we thank Prof. Zhilin Li (UNC Charlotte) for providing a code for thesolution of a Dirichlet problem for the Laplace equation by the immersed interface method, whichwe could use with some modifications for the solution of the direct problem in the anti-planarcase. Financial support is acknowledged to the Austrian National Science Foundation FWF undergrant SFB F 013 / 08 and P13478-INF.

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0.28

0.285

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0.295

0.3

0.305

0.31

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0.1 1

d H(Σ

δ α, Σ

∗

δ



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d 1(Σ

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Figure 9: Two Circle: Convergence in Hausdorff- and L1-distance versus noise δ

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